Optimal. Leaf size=484 \[ -\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{640 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \Pi \left (2;\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{128 a^3 d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A]
time = 1.03, antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2972, 3126,
3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} \frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{640 a^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{128 a^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 2972
Rule 3081
Rule 3126
Rule 3134
Rule 3138
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {3}{4} \left (32 a^2-5 b^2\right )+\frac {3}{2} a b \sin (c+d x)-\frac {5}{4} \left (16 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{20 a^2}\\ &=\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\int \csc ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {9}{8} b \left (36 a^2-5 b^2\right )-\frac {3}{4} a \left (16 a^2-b^2\right ) \sin (c+d x)-\frac {3}{8} b \left (128 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{60 a^2}\\ &=\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {3}{16} \left (128 a^4-116 a^2 b^2+15 b^4\right )-\frac {3}{8} a b \left (212 a^2+b^2\right ) \sin (c+d x)-\frac {3}{16} b^2 \left (404 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^2}\\ &=-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\int \frac {\csc (c+d x) \left (-\frac {45}{32} b \left (48 a^4+8 a^2 b^2-b^4\right )-\frac {3}{16} a b^2 \left (404 a^2-5 b^2\right ) \sin (c+d x)+\frac {3}{32} b \left (128 a^4-116 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^3}\\ &=-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}+\frac {\int \frac {\csc (c+d x) \left (\frac {45}{32} b^2 \left (48 a^4+8 a^2 b^2-b^4\right )+\frac {3}{32} a b \left (128 a^4+692 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{120 a^3 b}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{1280 a^3}\\ &=-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}+\frac {\left (3 b \left (48 a^4+8 a^2 b^2-b^4\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{256 a^3}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{1280 a^2}-\frac {\left (\left (128 a^4-116 a^2 b^2+15 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{1280 a^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{256 a^3 \sqrt {a+b \sin (c+d x)}}+\frac {\left (\left (128 a^4+692 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{1280 a^2 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{640 a^3 d}+\frac {3 b \left (36 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{320 a^2 d}+\frac {\left (32 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}-\frac {\left (128 a^4-116 a^2 b^2+15 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{640 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (128 a^4+692 a^2 b^2+5 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{640 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {3 b \left (48 a^4+8 a^2 b^2-b^4\right ) \Pi \left (2;\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{128 a^3 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 16.04, size = 544, normalized size = 1.12 \begin {gather*} \frac {-4 \cot (c+d x) \left (128 a^4-116 a^2 b^2+15 b^4-2 a b \left (236 a^2+5 b^2\right ) \csc (c+d x)+8 a^2 \left (-32 a^2+b^2\right ) \csc ^2(c+d x)+176 a^3 b \csc ^3(c+d x)+128 a^4 \csc ^4(c+d x)\right ) \sqrt {a+b \sin (c+d x)}+b \left (-\frac {2 i \left (128 a^4-116 a^2 b^2+15 b^4\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}-\frac {8 a b \left (404 a^2-5 b^2\right ) F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (1312 a^4+356 a^2 b^2-45 b^4\right ) \Pi \left (2;\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )}{2560 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2074\) vs.
\(2(545)=1090\).
time = 15.08, size = 2075, normalized size = 4.29
method | result | size |
default | \(\text {Expression too large to display}\) | \(2075\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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